Mirror object on x but also flip on Y?
The red object is not a reflection of the green object, wherever you put the mirror. It's a rotation of it. You could get there via 2 reflections, bit the mirror modifier would generate excess objects on the way.
An array modifier can do it using an axis/origin Empty with a rotation in Z of 180, the translation will depend on the origin of your object and the relative position of the Empty ...